Seating Arrangement

What the exam asks: five to eight persons sit in a straight line, and you must say who is at a slot, who is next to someone, or how many sit between two persons.

The logic behind it: a row is only a number line of slots 1 to n, so every clue turns into addition or subtraction on slot numbers.

The method: make the slots first, place the clues in the order of their strength, and verify at the end. After the basic row, this type covers three variations that sit on top of it: mixed facing, count clues, and sufficiency questions.

In a row facing North, the seats are slots 1 to n counted from the West end, which is the left end of the diagram. The whole puzzle is only a way of finding "which slot?". A clue gives an exact slot ("third from the left end"), a fixed gap ("only two persons between") or a neighbour ("immediately right of").

Draw n empty boxes first and number them. The slot from the left plus the slot from the right is always n + 1. So in a row of 7, "third from the right end" is slot 5.

[[fig:0]]

Now place the clues in order of strength. Exact slots go first. A glued pair (immediate right or left) moves as one block, so place it next. Gap clues come after that. The last person usually falls into the only empty slot.

Be careful with two phrases. "Immediately right of X" is the very next slot towards the East. "To the right of X" only means somewhere further East. And "second from one of the ends" gives two possible slots, 2 and n − 1. When a clue allows two slots, carry both until another clue kills one.

Some questions do not give the total. Then use the same slot arithmetic: if a person is 6th from the left and 4th from the right, the total is 6 + 4 − 1 = 9. Remember to take the person only once.

Sub-case A — Mixed facing in a row

In a row, a person's seat never changes when he turns round. Only his left and right swap. So a row with mixed facing is solved in two layers: first the positions, then the facing of each person.

A North-facing person has West on his left and East on his right. A South-facing person has East on his left and West on his right. Neighbours by position never change; only the side called left or right changes.

[[fig:1]]

In the figure, M faces South. His right hand points West (towards L) and his left hand points East (towards N). Everyone else faces North, so for them right is East.

Work like this. Write the positions in West-to-East order and put an arrow on every person, up for North and down for South. When a clue says "X sits to the immediate right of Y", ask first: does Y face North or South? If Y faces North, X is the next seat to the East. If Y faces South, X is the next seat to the West. The facing of X does not matter for this clue. Only the facing of the reference person, the one named after "of", counts.

Clues that only talk about neighbours, "between" or the ends need no facing at all. They behave exactly as in a normal row.

When the question mentions the ends, read it carefully. In a mixed row the setter usually names the West or the East end. If a question says "left end" without saying more, assume North-facing unless the question tells you otherwise.

Sub-case B — Count clues and an unknown total

Some clues do not say where a person sits. They compare counts: "As many persons sit to the right of Sameer as to the left of Radhika." Such a clue links two people through a number, and the number may be known, or unknown at first.

Take a row of n persons and a person at slot k (counted from the left). Persons to the left = k − 1. Persons to the right = n − k. Everything in this type is built from these two lines.

Use them like this. If Radhika has 4 persons on her left, her slot is 5. If Sameer has as many persons on his right as Radhika has on her left, then Sameer has 4 persons on his right, and his slot is n − 4. In the figure, R has 4 persons on her left, so she is at slot 5, and S has 6 persons on his right, so in a row of 9 he is at slot 3.

[[fig:2]]

There are two situations.

• n is given. Convert each count into a slot with the two lines above, then place the people. The answer is a simple difference of slots, minus 1 if the question asks for persons between.
• n is not given. Write the same count in two ways and equate them. For example, "persons to the right of X = persons to the left of Y" gives n − k(X) = k(Y) − 1. Place everyone else with the other clues, and solve for n. The final answer is n = (slots) + (persons on one side).

Persons between two seats = |difference of slots| − 1. Total persons = persons on the left + persons on the right + 1 for the person himself.

Keep the direction in mind: "left" and "right" in a row facing north are the usual west and east.

Sub-case C — Sufficiency of data [New Pattern]

In this form, the question is not "who sits where". The question asks whether the given statements are enough to answer it. You are given a question and two statements, and you must decide which statement, if any, answers the question with exactly one answer.

Think of it as three separate experiments.

1. Use Statement I alone. Try to draw the arrangement. If more than one arrangement is possible and they give different answers to the question, I alone is not sufficient.
2. Use Statement II alone, in the same way. Never carry facts from I into II.
3. If neither alone works, use both together. If the combined facts give a single answer, both together are sufficient. If even then two answers remain, the data is not sufficient.

Sufficient means one definite answer, not just an answer that is possible. If a statement leaves two arrangements and the two arrangements give the same answer, that statement is still sufficient. This is rare but true, so always check the answer, not just the number of arrangements.

Be careful with phrases that look precise but are not. "Second from an end" gives two cases, because the end can be left or right. In the figure, X is second from the left in the first row and second from the right in the second row. Unless the statement says which end, both cases stay alive.

[[fig:3]]

The same care is needed in a circle: "third to the right" in a circle of six is the opposite seat, and it is the same as "third to the left". In a row of seven, "somewhere to the left of" gives only a side.

The four options always read the same way: I alone, II alone, both together, or neither. Fix them in your mind, so that the mapping costs no time in the exam.

What the exam asks: six to eight persons sit around a table facing the centre. The clues say second or third to the left or right, opposite, between and neighbour, and the question asks who sits where, who sits opposite someone, or how many sit between two persons.

The logic behind it: a circle is a row whose two ends are glued together. If you number the seats clockwise, every clue becomes addition or subtraction on seat numbers, and you only wrap around at n.

The method: choose an anchor, number the seats, place the exact distances, then the opposite pairs, then the neighbours, and verify. A hexagon or pentagon table and an "interchange" sentence at the end of the story are the same circle; both are covered at the end of this type.

Around a circular table the seats have no ends, so we give them numbers. Call the seat of the anchor person seat 1 and count clockwise: 1, 2, 3 ... up to n. The anchor is the person who appears in the most clues.

Facing the centre, a person's left is clockwise and his right is anticlockwise. Third to the left is seat + 3; third to the right is seat − 3. In the figure, X sits at seat 1. The person on his right is at seat 8 (anticlockwise) and the person on his left is at seat 2.

[[fig:0]]

Opposite seats exist only when n is even, and then the opposite of seat i is seat i + n/2.

Handle "between" with care. "X sits between A and B and is adjacent to both" fixes X to the one seat that touches both. But "only X sits between A and B" can lie on the short arc or on the long arc, so keep both and kill one at the first contradiction. A clue such as "counting from the right of A, only two persons sit between A and B" fixes the arc: B is the third person to the right of A.

Work in this order. First number the seats and fix the anchor. Then place the clues with an exact distance ("second to the right of"). Next place the opposite pairs among the free seats. Then use the neighbour clues to settle the last names. Finally, write the seat number of the answer before you look for the name, and check every clue once.

Sub-case A — Polygon tables (triangle, pentagon, hexagon, octagon)

A regular polygon with one person at each corner is simply a circle with that many seats. Number the corners 1 to n clockwise, fix an anchor on corner 1, and use every concept of the circle. The shape adds nothing to the logic.

The only new question is the opposite corner. A hexagon (6 corners) and an octagon (8 corners) have opposite corners. In a hexagon the opposite of corner i is i + 3; in an octagon it is i + 4. A pentagon (5) and a triangle (3) have none, so no one can be opposite anyone there. Opposite exists only when the number of corners is even.

[[fig:1]]

The figure shows the hexagon with its three opposite pairs: A-D, B-E and C-F.

Sometimes persons sit at the corners and also at the middle of the sides. A triangle with persons at the three corners and at the three mid-sides has six seats, and corners and mid-sides alternate. Each corner is opposite the middle of the opposite side, so the opposite of seat i is i + 3 again.

[[fig:2]]

Facing is read from the story. Usually everyone faces the centre. A common variation says the corners face outside and the mid-side persons face the centre; then apply the Type 6 concept to each reference person.

For "not adjacent" clues in a hexagon, remember that every corner has two neighbours, two persons at second distance and one opposite. Use the "not adjacent" clue last, when two seats are left.

Sub-case B — Interchange (swap) questions

"Interchange" means that two persons exchange seats. Remember one sentence: the seats stay, the names move. The facing belongs to the seat as well. Unless the question says otherwise, the person who arrives in a seat takes over the facing of that seat, so in a circle that faces the centre nothing about left and right changes.

First solve the original arrangement completely. Then draw the new diagram by writing the two names in each other's seats. Only then read the new relation, using the usual facing concept. Do not work out new neighbours from the old ones in your head. Only the two swapped persons and the persons next to their seats get new neighbours; everybody else is untouched.

[[fig:3]]

In the figure, B and E interchange. The red person goes to the blue seat and the blue person goes to the red seat. After the swap, the neighbours of seat 2 are A and C as before, but the person between them is now E.

If there are two swaps, apply them one after the other in the order given, and redraw after each swap, or keep a small list of "seat → name" and change it step by step. The second swap works on the new arrangement, not on the original one.

A typical question is "After the interchange, who sits immediately right of X?". If X moved, use his new seat. If X did not move, his neighbour may have changed because the person next to him has moved.

What the exam asks: persons live on different floors of a building, or boxes are kept one over the other, and the question asks who is on a floor, how many are above or below someone, or how many are between two. The recent railway shifts repeat this story with a fresh set of clues, and the gaps are usually given as "only two people live between …".

The logic behind it: a building is a row standing upright, so every method of the Linear Row works with floor 1 at the bottom.

The method: pin the top or bottom person first, turn each gap into a pair of floors, then use the "below or above" chains. Clues that describe a floor by its number (even, odd, prime) and houses numbered 14, 16, 18 … are the same row in disguise; their special arithmetic follows.

In a building one person lives on each floor, and in a stack the boxes lie one above another. Both are a vertical line of slots, so the whole method is that of a row turned upright. Number the slots with 1 at the bottom, unless the question says that the top is 1.

"Above" is only a direction. "Immediately above" means that the floor number is exactly one more. "Exactly n persons between" means that the floor numbers differ by n + 1. In the figure, floors 2 and 5 have two floors between them, so they are 3 apart. Floors differ by (persons between) + 1.

[[fig:0]]

Place the clues in order of strength. First the exact floors (top floor, bottom floor, "lives on floor 3"). Then the glued pairs, such as "immediately above". Then the gap clues, and last the loose clues such as "somewhere above". For a gap clue, list all possible pairs. In a building of 7 floors, "exactly three between" means a difference of 4, so only the pairs (1, 5), (2, 6) and (3, 7) are possible. Some pairs fall off the building, which is the quickest way to kill a case.

A loose clue such as "X lives somewhere above Y" gives only an inequality. Keep it for the end, to choose between the two remaining options.

For boxes, "kept above" is the same as "above", and the box at the top is the last in the list. Read whether the question counts positions from the top or from the bottom, and write "1 = bottom" on the page to avoid mistakes.

Sub-case A — Even, odd and prime floors

Number properties cut the list of possible floors before any placing starts. For every person, write down the floors that his property allows. Only then use the gap and neighbour clues.

In an 8-floor building:

In a 9-floor building the primes are still 2, 3, 5, 7, and "neither even nor prime" leaves floors 1 and 9.

[[fig:1]]

In the figure, E marks even floors (blue), P marks prime floors (orange), and the floor 2 is both (purple). Floor 1 has no property at all.

Three facts cause most mistakes. One is not a prime number. Two is the only even prime. And floors start at 1: there is no floor 0, and no floor above the top.

"Neither even nor prime" means odd and not prime, so it is a short list. "Even but not the 6th" means even floors with floor 6 removed: 2, 4, 8 in an 8-floor building. Make the list first, then apply the gap.

Many questions give just two such properties and one gap clue, and the whole question finishes in three lines.

Sub-case B — Numbered houses

Houses on one side of a street carry numbers such as 14, 16, 18, 20, 22. This is only a row in disguise. The house number is a name for the slot, and the slot is what you really need.

Convert every number into a slot first. With the numbers 14, 16, 18, 20, 22, house 14 is slot 1, 16 is slot 2, and so on. Slot = (number − first number) ÷ step + 1. After this, the question is an ordinary row question.

[[fig:2]]

Now read the clue words carefully, because they mix two different counts.

• "The next higher number" means the next slot, one step along the street.
• "The next lower number" means the slot just before.
• "Greater than" and "less than" are only inequalities. They tell you which side, nothing more.
• "Exactly two houses between X and Y" counts houses, not numbers. The slots differ by 3, so the numbers differ by 3 steps. With a step of 2, that is a difference of 6 in the numbers.

In the figure, houses 14 and 22 have three houses between them (16, 18, 20), although the numbers differ by 8. Houses between = (difference of numbers ÷ step) − 1.

Work in this order. Fill the houses whose numbers are named in the clues. Then use "next higher or lower", then the gap clues, and keep "greater than" and "less than" for the last choice. A house that is greater than a placed house must lie beyond it, and this alone removes many cases.

If the numbers are odd, for example 31, 33, 35, the step is still 2. If the numbers run 101, 102, 103, the step is 1. Read the step from the list given in the question, never assume it.

What the exam asks: eight persons sit at the four corners and the four middles of the sides of a square table, and the question usually asks how many persons sit between two of them, counted from the left or the right of one. This form is a regular feature of the railway (NTPC) shifts.

The logic behind it: walk once round a square and you meet corner, side, corner, side, … So the table is an 8-seat circle with a code written on its seats: odd seats are corners and even seats are sides (or the other way round).

The method: straighten the table into a circle, use the corner or side code as a filter, then place the names as in the circle. A rectangular table is different: draw it and join the opposite pairs.

Around a square table, persons usually sit at the four corners and at the middle of the four sides: eight seats. Walk round the table clockwise from one corner and you meet corner, side, corner, side, and so on. So an eight-seat square table behaves exactly like an eight-seat circle. Number the seats 1 to 8 clockwise, starting at a corner.

[[fig:0]]

What is special about a square table is who faces where. Some questions say "everyone faces the centre". Others say "the persons at the corners face the centre and those at the middle of the sides face outside", or the other way round. Never assume the facing; read who faces in and who faces out. After that, use the circle concepts: facing in, right is anticlockwise; facing out, right is clockwise.

In the figure, corners face the centre and the middle-of-side seats face outside. The opposite of seat 1 is seat 5, the same as in an 8-seat circle: opposite of i is i + 4. A corner is opposite a corner and a mid-side is opposite a mid-side.

A clue "X sits at a corner" simply means X is on an odd seat if seat 1 is a corner. With four persons at corners and four at the middle of the sides, the two groups alternate around the table.

[[fig:1]]

A rectangular table needs more care because the long sides hold more seats than the short sides. With two seats on each long side and one at each end (six persons), the opposite seat is not i + 3. In the figure, A is opposite E, B is opposite D, and C is opposite F. Always draw a rectangular table and join the opposite pairs before you place a name.

What the exam asks: six, eight or ten persons sit in two parallel rows, each person facing someone in the other row. The questions ask who faces whom, who is second to the right of someone in a row, or which persons face north (or south). The same story is sometimes told with flats on a floor.

The logic behind it: the two rows face opposite ways, so the same word "right" means East in one row and West in the other. Once every left and right is converted into West and East, each row is an ordinary row, and "faces" only joins two equal slot numbers.

The method: draw both rows, convert, lock the "faces" pairs, then use the neighbour and end clues inside each row.

In a two-row arrangement the rows face each other. One row faces North and the other faces South. Each person of one row looks at exactly one person of the other row, so slot k of Row 1 faces slot k of Row 2 when both rows are counted from the West.

Because the rows face opposite ways, left and right point to opposite sides of the diagram. In the North-facing row right is East. In the South-facing row right is West. So the right end of Row 1 is the East end, but the right end of Row 2 is the West end, and the two right ends are not opposite each other. The right end of one row faces the left end of the other.

[[fig:0]]

In the figure, Row 1 (bottom) faces North and Row 2 (top) faces South. R and L show the right and left of each row. A faces E, B faces F and so on.

The method has four short steps. Draw both rows with slots 1 to 4 from West to East. Convert every left or right clue into West or East using the facing of that row. Use "A faces B" to lock two slots at once, because they must have the same slot number. Then use the row-end clues and neighbours inside a row.

Neighbour clues ("immediate right of", "second to the left of") always stay inside one row. Never move across to the other row with them.

Finally, check the facing of each row in the story. If both rows happen to face the same direction, left and right are the same side in both rows and the "facing each other" idea does not apply.

What the exam asks: a circle story with one extra line, "all face outside" or "some face the centre and some face outside", and then the usual clues of right, left, opposite and between. The question asks for the immediate left or right of a person, or for a neighbour.

The logic behind it: facing decides only which hand points clockwise. Facing the centre, right is anticlockwise; facing outside, right is clockwise. The order of the seats, and the words opposite, neighbour and between, do not change at all.

The method: put a small arrow, in or out, on every seat; use the arrow of the reference person (the one named after "of") for each left or right; never renumber the seats because of facing.

A person who faces outside has his hands on the opposite sides from a person who faces the centre. Facing outside, his right is clockwise and his left is anticlockwise. The order of persons around the circle never changes. Only the reading of left and right changes.

[[fig:0]]

In the figure, A faces the centre, so his right neighbour (R) is the anticlockwise one, F. D faces outside, so his right neighbour (R) is the clockwise one, E. The two arrows point to different seats even though the seats are numbered the same way.

The method is to mark a small arrow, in or out, on every seat first. Then, for the person you are asked about, or the person named as the reference in a clue, use the concept: inward means right is anticlockwise (seat − k); outward means right is clockwise (seat + k). Never change the seat numbering because of the facing.

If everybody faces outside, solve the puzzle exactly like a Type 2 circle, reading every clue and the question in the same words. Your drawing will be the mirror image of the real seating, but every question has the same answer. This works only when all the persons face the same way.

In a mixed circle, the facing that counts in a clue is the facing of the reference person, the one named after "of". The clue "X sits second to the left of Y" uses Y's facing. Opposite, adjacent and between do not depend on facing; only left and right do.

What the exam asks: a circle story with extra tags on the persons, such as gender, married couples or professions. Older railway shifts (NTPC and JE) used it, and the same pattern also comes as a vertical row of nine where boys and girls stand alternately.

The logic behind it: the tags fix the pattern of the seats before any name is placed. Alternate genders give men the odd seats and women the even seats; "every man sits opposite his wife" gives fixed opposite pairs.

The method: pattern first, position clues second, professions last.

Extra labels such as gender, marriage or profession restrict the seats. Treat each label as a second tag on the same seat, and use it to fix the pattern of seats before any name is placed.

The strongest pattern is alternate gender. If men and women sit alternately, men take every other seat. Put the men on the odd seats and the women on the even seats (or the reverse). The seats for each group are then known before you read a single name clue, which cuts the work by a large factor.

For couples, "every man sits opposite his wife" in a six-seat circle gives the pairs (1, 4), (2, 5), (3, 6). Notice that opposite seats in a six-seat circle are three apart, so one is odd and the other even, and this agrees with alternate genders.

[[fig:0]]

Test whether the concepts can hold together before you start. In an eight-seat circle opposite seats are four apart and have the same parity, so "men and women alternate" and "every man sits opposite his wife" cannot both be true there. If a puzzle states such a pair, one of the statements must mean something else, such as "adjacent".

Designations work the same way. Make a small table seat → name → post. Fill the seats first with the position clues, and match the posts to the names only at the end.

A question such as "Who is X's wife?" has a direct answer when couples are opposite: it is the person opposite X. When couples are named in the story, the question is only about seats.